Problem: $ B = \left[\begin{array}{rr}-2 & -2 \\ -1 & -1 \\ 5 & 3\end{array}\right]$ $ F = \left[\begin{array}{rr}4 & 2 \\ 5 & -1\end{array}\right]$ What is $ B F$ ?
Solution: Because $ B$ has dimensions $(3\times2)$ and $ F$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ B F = \left[\begin{array}{rr}{-2} & {-2} \\ {-1} & {-1} \\ \color{gray}{5} & \color{gray}{3}\end{array}\right] \left[\begin{array}{rr}{4} & \color{#DF0030}{2} \\ {5} & \color{#DF0030}{-1}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ B$ , with the corresponding elements in column $j$ of the second matrix, $ F$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ B$ with the first element in ${\text{column }1}$ of $ F$ , then multiply the second element in ${\text{row }1}$ of $ B$ with the second element in ${\text{column }1}$ of $ F$ , and so on. Add the products together. $ \left[\begin{array}{rr}{-2}\cdot{4}+{-2}\cdot{5} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ B$ with the corresponding elements in ${\text{column }1}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{-2}\cdot{4}+{-2}\cdot{5} & ? \\ {-1}\cdot{4}+{-1}\cdot{5} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ B$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{-2}\cdot{4}+{-2}\cdot{5} & {-2}\cdot\color{#DF0030}{2}+{-2}\cdot\color{#DF0030}{-1} \\ {-1}\cdot{4}+{-1}\cdot{5} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{-2}\cdot{4}+{-2}\cdot{5} & {-2}\cdot\color{#DF0030}{2}+{-2}\cdot\color{#DF0030}{-1} \\ {-1}\cdot{4}+{-1}\cdot{5} & {-1}\cdot\color{#DF0030}{2}+{-1}\cdot\color{#DF0030}{-1} \\ \color{gray}{5}\cdot{4}+\color{gray}{3}\cdot{5} & \color{gray}{5}\cdot\color{#DF0030}{2}+\color{gray}{3}\cdot\color{#DF0030}{-1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-18 & -2 \\ -9 & -1 \\ 35 & 7\end{array}\right] $